A Theoretical Omega-Square Model Considering the Spatial Variation in Slip and Rupture Velocity

Abstract

A theoretical model for constructing the x-squared model is proposed by modifying the k-squared model of Bernard et al. (1996). The k-squared model provides a theoretical basis for the empirical x-squared model under the assumptions that (1) the spatial wavenumber spectrum of the slip distribution falls off as the inverse of the wavenumber squared (k-squared), (2) the Fourier amplitudes of the slip velocity are independent of x at high frequencies, and (3) the rupture velocity is constant. In this study, a more realistic model is proposed by modifying the last two assumptions. First, a Kostrov-type slip velocity model is proposed by superposing equilateral triangles, in which a source-controlled fmax is imposed by the minimum duration among the triangles. The Fourier amplitude of our slip velocity model falls off as the inverse of x at high frequencies less than fmax. Next, in order to model variable rupture velocities, the incoherent rupture time (Dt r), namely, the difference between the actual rupture time and the coherent (average) rupture time, is introduced. After checking various models for Dt r distributions, the k-squared model for Dt r , similar to that for the slip distributions of the k-squared model, is found to be the most plausible. Finally, it is confirmed that the proposed source model (we call it as the x-inverse-squared model), which consists of the combination of the slip velocity proposed here and the k-squared distributions for both slip and Dt r , not only is consistent with the empirical x-squared model, but also provides the theoretical basis for constructing realistic source models at broadband frequencies.

Implications of the 26 December 2004 Sumatra–Andaman Earthquake on Tsunami Forecast and Assessment Models for Great Subduction-Zone Earthquakes

“…Tsunami generation models used for tsunami assessments such as presented here have been modified from models previously used for strong ground motion studies (Herrero and Bernard, 1994;Berge et al, 1998;Somerville et al, 1999;Hisada, 2000Hisada, , 2001Honda and Yomogida, 2003). At low wavenumbers, stochastic slip distributions are scaled relative to average slip or static stress drop.…”

“…Each of the slip distributions conform to a k ‫2מ‬ radial-wavenumber spectrum for k Ͼ k c where k c scales with the characteristic rupture dimension (Herrero and Bernard, 1994;Tsai, 1997;Somerville et al, 1999;Hisada, 2000Hisada, , 2001Mai and Beroza, 2002). For the more general case of a k ‫␣מ‬ spectrum, Zeng et al (2005) note that scaling of ū with rupture length (L) is physically related to the degree of slip heterogeneity (␣) and indicate that linear scaling between ū and L occurs only for relatively smooth slip distributions.…”

Simulation of Ground Motion Using the Stochastic Method

“…Furthermore, the effects of a finite-fault averaged over a number of sites distributed around the fault (to average over radiation pattern and directivity effects) can be captured in several ways: 1) using the closest distance to faulting (as is done in empirically derived ground-motion prediction equations) as the source-to-site distance; 2) using a two-corner source spectrum (ATKINSON and SILVA, 2000); 3) allowing the geometrical spreading to be magnitude dependent . In addition, it should be possible to extend the method to account for specific fault-station geometries in a simple way, perhaps combining the simple computation of envelopes of acceleration (MIDORIKAWA and KOBAYASHI, 1978;COCCO and BOATWRIGHT, 1993) with statistical descriptions of the source (e.g., LOMNITZ-ADLER and LUND, 1992;HERRERO and BERNARD, 1994;JOYNER, 1995;BERNARD et al, 1996;HISADA, 2000). The overriding philosophy of such an effort would be to capture the essence of motions from an extended rupture without sacrificing the conceptual simplicity of the stochastic method.…”

Effect of complex fault geometry and slip style on near-fault strong motions and static displacement

“…Synthesized seismograms obtained from a simple Haskell model show good agreement with observed data in the low-frequency range, but it is still very difficult to explain observed high-frequency waves, such as those in a frequency range higher than 1 Hz, in a deterministic manner. To reproduce a spectrum of seismic body-waves (e.g., the ω-square model) at high frequency, several studies have introduced kinematic stochastic models (e.g., Herrero and Bernard, 1994;Bernard et al, 1996;Hisada, 2000) considering various rupture velocities and slip distributions on a fault.…”